Finite Group Databasebeta

← #14326829

Subgroups — #14326829

Order 1920.

1008 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91357C211yesmaximal, characteristic
2#11750749C211yesmaximal, characteristic, complemented
3C5×#4648C211yesmaximal, characteristic, complemented
4C3×#64955C211yesmaximal, characteristic
5#11748867C211yesmaximal, characteristic, complemented
6C4×#6668C212yesmaximal, complemented
7#11748802C212yesmaximal, complemented
8#11748738C211yesmaximal, characteristic
9#11748710C212yesmaximal
10#11748705C212yesmaximal
11#11748600C211yesmaximal, characteristic, complemented
12#10605123—31nomaximal, Hall
13C22×C3×C4×D5C2211yescharacteristic
14C2×C4×Dic15C2212yes—
15C2×C4×Dic15C2211yescharacteristic
16C2×C4×C5×Dic3C2212yes—
17C2×C4×C5×Dic3C2211yescharacteristic
18C2×C3×#685C2212yes—
19C2×C3×C4×Dic5C2212yes—
20C2×Dic3⋊Dic5C2212yes—
21C2×#6669C2212yes—
22C2×#6668C2212yes—
23C2×#6668C2211yescharacteristic
24C2×#6668C2211yescharacteristic
25C2×#6665C2212yes—
26C2×#6665C2211yescharacteristic
27C2×#6665C2211yescharacteristic
28#91357C418yescomplemented
29C3×C5×#95C2211yescharacteristic
30#91115C2212yes—
31#91115C2211yescharacteristic
32C5×C3⋊#95C2211yescharacteristic
33C5×C3⋊#95C2212yes—
34C3×#3963C2211yescharacteristic
35#90994C2212yes—
36#90994C2212yes—
37#73459—51nomaximal, Hall
38C2×#4018—31no—
39C5×#376—31no—
40#65229—32no—
41#65225—31no—
42C4×#683—32no—
43#64959—32no—
44#64956—31no—
45#64955D311yescharacteristic
46#64939—31no—
47#64937—31no—
48#64936—32no—
49C23×C3×D5C2311yescharacteristic
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2312yes—
52C22×Dic15C2312yes—
53C22×Dic15C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×C5×Dic3C2312yes—
56C22×C5×Dic3C2311yescharacteristic
57C22×C5×Dic3C2311yescharacteristic
58C22×C3×Dic5C2312yes—
59C22×C3×Dic5C2311yescharacteristic
60C2×C3×C4×D5—24no—
61C2×C3×C4×D5C2×C414yes—
62C4×Dic15—24no—
63C4×Dic15C2×C414yes—
64C4×C5×Dic3—24no—
65C4×C5×Dic3C2×C414yes—
66C3×#685—24no—
67C3×C4×Dic5—24no—
68Dic3⋊Dic5—42no—
69#6669—42no—
70#6668—24no—
71#6668C2×C414yes—
72#6668C2×C414yes—
73#6665—42no—
74#6665C2×C414yes—
75#6665C2×C414yes—
76C2×#926—51no—
77C3×C4×C22⋊C4—51no—
78#5212—51no—
79#4944—52no—
80#4944—52no—
81C4×C4⋊Dic3—52no—
82#4928—52no—
83#4650—51no—
84#4650—51no—
85#4648—51no—
86#4648D511yescharacteristic, complemented
87C22×C4×D5C2×D311yescharacteristic
88C2×C42×C5—31no—
89C2×C42×C5—32no—
90C2×#685—31no—
91C2×#685—31no—
92C2×#685—32no—
93C2×#685C2×D312yes—
94C2×C4⋊Dic5—32no—
95C2×#683—31no—
96C2×#683—31no—
97C2×#683—32no—
98C2×#683—32no—
99C2×C4×Dic5—31no—
100C2×C4×Dic5—32no—
101C2×C4×Dic5C2×D312yes—
102#4018—38no—
103C5×#95—32no—
104C5×#95—31no—
105C5×#95C2×D311yescharacteristic
106#3963—32no—
107#3963—31no—
108#3963—32no—
109#3963—32no—
110#3963C2×D311yescharacteristic
111#2346—151noSylow
112C23×C3×C5C2411yescharacteristic
113C22×C3×D5—22no—
114C22×C3×D5—22no—
115C22×C3×D5—21no—
116C22×C3×D5—21no—
117C22×C3×D5C22×C412yes—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5C22×C412yes—
122C2×C3×C4×C5#4114yes—
123C2×Dic15—22no—
124C2×Dic15—21no—
125C2×Dic15—21no—
126C2×Dic15—22no—
127C2×Dic15—22no—
128C2×Dic15—22no—
129C2×Dic15C22×C412yes—
130C2×Dic15C22×C412yes—
131C2×C5×Dic3—22no—
132C2×C5×Dic3—21no—
133C2×C5×Dic3—21no—
134C2×C5×Dic3—22no—
135C2×C5×Dic3—22no—
136C2×C5×Dic3—22no—
137C2×C5×Dic3C22×C412yes—
138C2×C5×Dic3C22×C412yes—
139C2×C3×Dic5—22no—
140C2×C3×Dic5—22no—
141C2×C3×Dic5—22no—
142C2×C3×Dic5C22×C412yes—
143C2×C3×Dic5#4114yes—
144C3×C4×D5—28no—
145C23×C3×C4—51no—
146C2×C3×C22⋊C4—52no—
147C2×C3×C42—52no—
148C2×#217—51no—
149C2×#217—51no—
150C2×#217—52no—
151#926—58no—
152C2×C4⋊Dic3—51no—
153C2×C4⋊Dic3—51no—
154C2×C4⋊Dic3—52no—
155C2×Dic3⋊C4—52no—
156C2×Dic3⋊C4—52no—
157C2×C4×Dic3—51no—
158C2×C4×Dic3—52no—
159C2×C4×Dic3C2×D512yes—
160C2×C4×Dic3C2×D511yescharacteristic
161C3×#95—51no—
162C3×#95C2×D511yescharacteristic
163C3⋊#95—51no—
164C3⋊#95—52no—
165C3⋊#95—52no—
166C3⋊#95—52no—
167C3⋊#95C2×D511yescharacteristic
168C3⋊#95C2×D512yes—
169C23×D5C22×D311yescharacteristic
170C22×C4×C5—31no—
171C22×C4×C5—31no—
172C22×C4×C5—32no—
173C22×C4×C5C22×D311yescharacteristic
174C22×C4×C5C22×D312yes—
175C22×Dic5—31no—
176C22×Dic5—31no—
177C22×Dic5—32no—
178C22×Dic5C22×D312yes—
179C22×Dic5C22×D311yescharacteristic
180C2×C4×D5—24no—
181C2×C4×D5C4×D314yes—
182C42×C5—34no—
183C42×C5—64no—
184#685—24no—
185#685—34no—
186#685—34no—
187#685—122no—
188C4⋊Dic5—122no—
189#683—34no—
190#683—34no—
191#683—64no—
192#683—122no—
193C4×Dic5—24no—
194C4×Dic5—34no—
195C4×Dic5—64no—
196C2×#117—151no—
197#389—152no—
198#388—152no—
199#388—152no—
200#382—151no—
201#382—151no—
202#380—151no—
203C4×C4⋊C4—152no—
204C4×C22⋊C4—51no—
205#376—31no—
206#376—151no—
207C22×C3×C5C23×C411yescharacteristic
208C22×C3×C5C2×#4112yes—
209C22×C3×C5C2×#4112yes—
210C22×C3×C5H(5,𝔽2)11yescharacteristic
211C22×C3×C52−1+411yescharacteristic
212C2×C3×D5—22no—
213C2×C3×D5—22no—
214C2×C3×D5—22no—
215C2×C3×D5—42no—
216C2×C3×D5—42no—
217C3×C4×C5—22no—
218C3×C4×C5—24no—
219Dic15—22no—
220Dic15—22no—
221Dic15—42no—
222C3×Dic5—42no—
223C3×Dic5#11714yes—
224C5×Dic3—22no—
225C5×Dic3—22no—
226C5×Dic3—42no—
227C24×C3—51no—
228C22×C3×C4—51no—
229C22×C3×C4—54no—
230C22×C3×C4—52no—
231C22×C3×C4—104no—
232C22×C3×C4C22×D511yescharacteristic
233C22×C3×C4C22×D512yes—
234C22×Dic3—51no—
235C22×Dic3—51no—
236C22×Dic3—52no—
237C22×Dic3C22×D512yes—
238C22×Dic3C22×D511yescharacteristic
239C22×Dic3C22×D511yescharacteristic
240C3×C22⋊C4—104no—
241C3×C42—104no—
242#217—54no—
243#217—54no—
244#217—202no—
245C4⋊Dic3—54no—
246C4⋊Dic3—54no—
247C4⋊Dic3—104no—
248Dic3⋊C4—202no—
249Dic3⋊C4—202no—
250C4×Dic3—24no—
251C4×Dic3—54no—
252C4×Dic3—104no—
253C4×Dic3C4×D514yes—
254C23×C5C23×D311yescharacteristic
255C22×D5—22no—
256C22×D5—22no—
257C22×D5—21no—
258C22×D5—21no—
259C22×D5C2×C4×D312yes—
260C2×C4×C5—22no—
261C2×C4×C5—22no—
262C2×C4×C5—22no—
263C2×C4×C5—32no—
264C2×C4×C5—32no—
265C2×C4×C5—62no—
266C2×C4×C5—61no—
267C2×C4×C5—61no—
268C2×C4×C5—62no—
269C2×C4×C5—62no—
270C2×C4×C5—62no—
271C2×C4×C5C2×C4×D312yes—
272C2×C4×C5#23514yes—
273C2×Dic5—22no—
274C2×Dic5—22no—
275C2×Dic5—22no—
276C2×Dic5—32no—
277C2×Dic5—32no—
278C2×Dic5—62no—
279C2×Dic5—61no—
280C2×Dic5—61no—
281C2×Dic5—62no—
282C2×Dic5—62no—
283C2×Dic5—62no—
284C2×Dic5C2×C4×D312yes—
285C2×Dic5#23714yes—
286C4×D5—28no—
287C23×C4—51no—
288#117—158no—
289C2×C4⋊C4—151no—
290C2×C4⋊C4—151no—
291C2×C4⋊C4—152no—
292C2×C4⋊C4—152no—
293C2×C4⋊C4—152no—
294C2×C22⋊C4—52no—
295C2×C22⋊C4—151no—
296C2×C22⋊C4—151no—
297C2×C22⋊C4—152no—
298C2×C42—31no—
299C2×C42—32no—
300C2×C42—52no—
301C2×C42—151no—
302C2×C42—152no—
303#95—31no—
304#95—32no—
305#95—51no—
306#95—152no—
307#95—152no—
308#95—151no—
309#95—152no—
310#95D3×D511yescharacteristic
311C2×C3×C5C2×#11711yescharacteristic
312C2×C3×C5C4×#4111yescharacteristic
313C2×C3×C5#52011yescharacteristic
314C2×C3×C5#55212yes—
315C2×C3×C5#55512yes—
316C3×D5—42no—
317C23×C3—52no—
318C23×C3—102no—
319C23×C3—102no—
320C23×C3—101no—
321C23×C3—101no—
322C23×C3C23×D511yescharacteristic
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—52no—
327C2×C3×C4—54no—
328C2×C3×C4—108no—
329C2×C3×C4—102no—
330C2×C3×C4—102no—
331C2×C3×C4—102no—
332C2×C3×C4C2×C4×D512yes—
333C2×C3×C4#70914yes—
334C2×Dic3—22no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—52no—
341C2×Dic3—52no—
342C2×Dic3—102no—
343C2×Dic3—101no—
344C2×Dic3—101no—
345C2×Dic3—102no—
346C2×Dic3—102no—
347C2×Dic3—102no—
348C2×Dic3C2×C4×D512yes—
349C2×Dic3C2×C4×D512yes—
350C22×C5C22×C4×D311yescharacteristic
351C22×C5C2×#23512yes—
352C22×C5C2×#23712yes—
353C22×C5#100911yescharacteristic
354C22×C5#101011yescharacteristic
355C2×D5—22no—
356C2×D5—22no—
357C2×D5—22no—
358C2×D5—42no—
359C2×D5—42no—
360C4×C5—22no—
361C4×C5—24no—
362C4×C5—62no—
363C4×C5—62no—
364C4×C5—122no—
365Dic5—42no—
366Dic5—62no—
367Dic5—62no—
368Dic5—122no—
369Dic5#87714yes—
370C24—51no—
371C22×C4—31no—
372C22×C4—31no—
373C22×C4—32no—
374C22×C4—51no—
375C22×C4—54no—
376C22×C4—52no—
377C22×C4—104no—
378C22×C4—151no—
379C22×C4—151no—
380C22×C4—152no—
381C22×C4C2×D3×D512yes—
382C22×C4C2×D3×D511yescharacteristic
383C4⋊C4—154no—
384C4⋊C4—154no—
385C4⋊C4—304no—
386C4⋊C4—602no—
387C4⋊C4—602no—
388C22⋊C4—104no—
389C22⋊C4—154no—
390C22⋊C4—154no—
391C22⋊C4—602no—
392C42—34no—
393C42—64no—
394C42—104no—
395C42—154no—
396C42—304no—
397C3×C5#234611yescharacteristic, complemented, Hall
398C22×C3—102no—
399C22×C3—102no—
400C22×C3—102no—
401C22×C3—202no—
402C22×C3—202no—
403C22×C3C22×C4×D511yescharacteristic
404C22×C3C2×#70912yes—
405C22×C3D5×#4112yes—
406C22×C3#414911yescharacteristic
407C22×C3#415011yescharacteristic
408C3×C4—22no—
409C3×C4—24no—
410C3×C4—54no—
411C3×C4—202no—
412Dic3—22no—
413Dic3—22no—
414Dic3—42no—
415Dic3—102no—
416Dic3—102no—
417Dic3—202no—
418C2×C5C4×#23511yescharacteristic
419C2×C5C2×#87711yescharacteristic
420C2×C5#552311yescharacteristic
421C2×C5#554112yes—
422C2×C5#556312yes—
423D5—42no—
424C23—52no—
425C23—102no—
426C23—101no—
427C23—102no—
428C23—101no—
429C23C22×D3×D511yescharacteristic
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—32no—
434C2×C4—32no—
435C2×C4—52no—
436C2×C4—54no—
437C2×C4—62no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—62no—
443C2×C4—108no—
444C2×C4—102no—
445C2×C4—102no—
446C2×C4—102no—
447C2×C4—152no—
448C2×C4—152no—
449C2×C4—302no—
450C2×C4—301no—
451C2×C4—301no—
452C2×C4—302no—
453C2×C4—302no—
454C2×C4—302no—
455C2×C4#677114yes—
456C2×C4C4×D3×D512yes—
457C2×C3—202no—
458C2×C3C4×#70911yescharacteristic
459C2×C3D5×#11711yescharacteristic
460C2×C3#6585711yescharacteristic
461C2×C3#6586512yes—
462C2×C3#6586612yes—
463C5#7345911yescharacteristic, complemented, Sylow
464C22—102no—
465C22—102no—
466C22—102no—
467C22—202no—
468C22—202no—
469C22C2×#677112yes—
470C22C2×C4×D3×D511yescharacteristic
471C22#9202011yescharacteristic
472C22D5×#23712yes—
473C22#9203511yescharacteristic
474C4—24no—
475C4—22no—
476C4—54no—
477C4—62no—
478C4—62no—
479C4—122no—
480C4—202no—
481C4—302no—
482C4—302no—
483C4—602no—
484C3#1060512311yescharacteristic, complemented, Sylow
485C2—202no—
486C2C4×#677111yescharacteristic
487C2#1175280112yes—
488C2D5×#87711yescharacteristic
489C2#1175367011yescharacteristic
490C2#1175373612yes—
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC3×C5×#95C22
socle, cosocleC23×C3×C5C24
prime radicalC23×C3×D5C23
hypercenter#95D3×D5
hypocenterC3×C5#2346
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #95

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11750749C2semidirect
3. C5×#4648C2semidirect
5. #11748867C2semidirect
6. C4×#6668C2semidirect
7. #11748802C2semidirect
11. #11748600C2semidirect
28. #91357C4semidirect
86. #4648D5semidirect
397. C3×C5#2346semidirect
463. C5#73459semidirect
484. C3#10605123semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#913572
#117507492
C5×#46482
C3×#649552
#117488672
C4×#66682
#117488022
#117487382
#117487102
#117487052
#117486002
#106051233
#734595